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Ramanujan Graphs of Every Degree Adam Marcus (Crisply, Yale) Daniel Spielman (Yale) Nikhil Srivastava (MSR India)

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Expander Graphs Sparse, regular well-connected graphs with many properties of random graphs. Random walks mix quickly. Every small set of vertices has many neighbors. Pseudo-random generators. Error-correcting codes. Sparse approximations of complete graphs. Major theorems in Theoretical Computer Science.

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Spectral Expanders Let G be a graph and A be its adjacency matrix a c d e b 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0 Eigenvalues If d -regular (every vertex has d edges), trivial

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Spectral Expanders If bipartite (all edges between two parts/colors) eigenvalues are symmetric about 0 trivial a cd f b e If d -regular and bipartite, 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0

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Spectral Expanders 0 [] -d d G is a good spectral expander if all non-trivial eigenvalues are small

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Bipartite Complete Graph Adjacency matrix has rank 2, so all non-trivial eigenvalues are 0 a cd f b e 0 0 0 1 1 1 1 1 1 0 0 0

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Spectral Expanders 0 [] -d d Challenge: construct infinite families of fixed degree G is a good spectral expander if all non-trivial eigenvalues are small

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Spectral Expanders G is a good spectral expander if all non-trivial eigenvalues are small 0 [] -d d Challenge: construct infinite families of fixed degree Alon-Boppana 86: Cannot beat ) (

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G is a Ramanujan Graph if absolute value of non-trivial eigs 0 [] -d d ) ( Ramanujan Graphs:

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G is a Ramanujan Graph if absolute value of non-trivial eigs 0 [] -d d ) ( Ramanujan Graphs: Margulis, Lubotzky-Phillips-Sarnak88: Infinite sequences of Ramanujan graphs exist for

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G is a Ramanujan Graph if absolute value of non-trivial eigs 0 [] -d d ) ( Ramanujan Graphs: Margulis, Lubotzky-Phillips-Sarnak88: Infinite sequences of Ramanujan graphs exist for Can be quickly constructed: can compute neighbors of a vertex from its name

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G is a Ramanujan Graph if absolute value of non-trivial eigs 0 [] -d d ) ( Ramanujan Graphs: Friedman08: A random d -regular graph is almost Ramanujan :

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Ramanujan Graphs of Every Degree Theorem: there are infinite families of bipartite Ramanujan graphs of every degree.

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Theorem: there are infinite families of bipartite Ramanujan graphs of every degree. And, are infinite families of (c,d) -biregular Ramanujan graphs, having non-trivial eigs bounded by Ramanujan Graphs of Every Degree

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Find an operation that doubles the size of a graph without creating large eigenvalues. 0 [] -d d Bilu-Linial 06 Approach ( )

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0 [] -d d Bilu-Linial 06 Approach Find an operation that doubles the size of a graph without creating large eigenvalues. ( )

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2-lifts of graphs a c d e b

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a c d e b a c d e b duplicate every vertex

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2-lifts of graphs duplicate every vertex a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 c1c1 d1d1 e1e1 b1b1

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2-lifts of graphs a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1 c1c1 for every pair of edges: leave on either side (parallel), or make both cross

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2-lifts of graphs for every pair of edges: leave on either side (parallel), or make both cross a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1 c1c1

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2-lifts of graphs 0 1 0 0 1 1 0 1 0 1 0 1 0 1 0 0 0 1 0 1 1 1 0 1 0

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2-lifts of graphs 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0

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2-lifts of graphs 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0

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Eigenvalues of 2-lifts (Bilu-Linial) Given a 2-lift of G, create a signed adjacency matrix A s with a -1 for crossing edges and a 1 for parallel edges 0 -1 0 0 1 -1 0 1 0 1 0 1 0 -1 0 0 0 -1 0 1 1 1 0 1 0 a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1 c1c1

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Eigenvalues of 2-lifts (Bilu-Linial) Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of A s (new) 0 -1 0 0 1 -1 0 1 0 1 0 1 0 -1 0 0 0 -1 0 1 1 1 0 1 0 a0a0 c0c0 d0d0 e0e0 b0b0 a1a1 d1d1 e1e1 b1b1 c1c1

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Eigenvalues of 2-lifts (Bilu-Linial) Conjecture: Every d -regular graph has a 2-lift in which all the new eigenvalues have absolute value at most Theorem: The eigenvalues of the 2-lift are the union of the eigenvalues of A (old) and the eigenvalues of A s (new)

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Eigenvalues of 2-lifts (Bilu-Linial) Conjecture: Every d -regular graph has a 2-lift in which all the new eigenvalues have absolute value at most Would give infinite families of Ramanujan Graphs: start with the complete graph, and keep lifting.

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Eigenvalues of 2-lifts (Bilu-Linial) Conjecture: Every d -regular graph has a 2-lift in which all the new eigenvalues have absolute value at most We prove this in the bipartite case. a 2-lift of a bipartite graph is bipartite

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Eigenvalues of 2-lifts (Bilu-Linial) Theorem: Every d -regular graph has a 2-lift in which all the new eigenvalues have absolute value at most Trick: eigenvalues of bipartite graphs are symmetric about 0, so only need to bound largest

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Eigenvalues of 2-lifts (Bilu-Linial) Theorem: Every d -regular bipartite graph has a 2-lift in which all the new eigenvalues have absolute value at most

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First idea: a random 2-lift Specify a lift by Pick s uniformly at random

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First idea: a random 2-lift Specify a lift by Pick s uniformly at random Are graphs for which this usually fails

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First idea: a random 2-lift Specify a lift by Pick s uniformly at random Are graphs for which this usually fails Bilu and Linial proved G almost Ramanujan, implies new eigenvalues usually small. Improved by Puder and Agarwal-Kolla-Madan

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The expected polynomial Consider

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The expected polynomial Consider Conclude there is an s so that Prove Prove is an interlacing family

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The expected polynomial Theorem (Godsil-Gutman 81): the matching polynomial of G

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The matching polynomial (Heilmann-Lieb 72) m i = the number of matchings with i edges

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one matching with 0 edges

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7 matchings with 1 edge

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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0 0 1 x 1 1 0 0 0 1 x Expand using permutations

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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0 0 1 x 1 1 0 0 0 1 x Expand using permutations same edge: same value

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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0 0 1 x 1 1 0 0 0 1 x Expand using permutations same edge: same value

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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0 0 1 x 1 1 0 0 0 1 x Expand using permutations Get 0 if hit any 0s

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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0 0 1 x 1 1 0 0 0 1 x Expand using permutations Get 0 if take just one entry for any edge

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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0 0 1 x 1 1 0 0 0 1 x Expand using permutations Only permutations that count are involutions

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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0 0 1 x 1 1 0 0 0 1 x Expand using permutations Only permutations that count are involutions

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Proof that x 1 0 0 1 1 1 x 1 0 0 0 0 1 x 1 0 0 0 0 1 x 1 0 1 0 0 1 x 1 1 0 0 0 1 x Expand using permutations Only permutations that count are involutions Correspond to matchings

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The matching polynomial (Heilmann-Lieb 72) Theorem (Heilmann-Lieb) all the roots are real

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The matching polynomial (Heilmann-Lieb 72) Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most

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The matching polynomial (Heilmann-Lieb 72) Theorem (Heilmann-Lieb) all the roots are real and have absolute value at most Implies

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Interlacing Polynomial interlaces if

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Common Interlacing and have a common interlacing if can partition the line into intervals so that each interval contains one root from each poly

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Common Interlacing and have a common interlacing if can partition the line into intervals so that each interval contains one root from each poly ) ) ) ) ( (( (

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If p 1 and p 2 have a common interlacing, for some i. Largest root of average Common Interlacing

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If p 1 and p 2 have a common interlacing, for some i. Largest root of average Common Interlacing

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is an interlacing family If the polynomials can be placed on the leaves of a tree so that when put average of descendants at nodes siblings have common interlacings Interlacing Family of Polynomials

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is an interlacing family Interlacing Family of Polynomials If the polynomials can be placed on the leaves of a tree so that when put average of descendants at nodes siblings have common interlacings

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Interlacing Family of Polynomials There is an s so that Theorem:

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An interlacing family Theorem: Let is an interlacing family

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Interlacing and have a common interlacing iff is real rooted for all

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To prove interlacing family Let

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To prove interlacing family Let Need to prove that for all, is real rooted

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To prove interlacing family Let are fixed is 1 with probability, -1 with are uniformly Need to prove that for all, is real rooted

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Generalization of Heilmann-Lieb We prove is real rooted for every independent distribution on the entries of s

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Generalization of Heilmann-Lieb We prove is real rooted for every independent distribution on the entries of s By using mixed characteristic polynomials

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Mixed Characteristic Polynomials For independently chosen random vectors is their mixed characteristic polynomial. Theorem: Mixed characteristic polynomials are real rooted. Proof: Using theory of real stable polynomials.

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Mixed Characteristic Polynomials For independently chosen random vectors is their mixed characteristic polynomial. Obstacle: our matrix is a sum of random rank-2 matrices 0 -1 -1 0 0 1 1 0 or

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Mixed Characteristic Polynomials For independently chosen random vectors is their mixed characteristic polynomial. Solution: add to the diagonal 1 -1 -1 1 1 1 or

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Generalization of Heilmann-Lieb We prove is real rooted for every independent distribution on the entries of s Implies is an interlacing family

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Generalization of Heilmann-Lieb We prove is real rooted for every independent distribution on the entries of s Conclude there is an s so that Implies is an interlacing family

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Universal Covers The universal cover of a graph G is a tree T of which G is a quotient. vertices map to vertices edges map to edges homomorphism on neighborhoods Is the tree of non-backtracking walks in G. The universal cover of a d -regular graph is the infinite d -regular tree.

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Quotients of Trees d -regular Ramanujan as quotient of infinite d -ary tree Spectral radius and norm of inf d -ary tree are

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Godsils Proof of Heilmann-Lieb T(G,v) : the path tree of G at v vertices are paths in G starting at v edges to paths differing in one step

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Godsils Proof of Heilmann-Lieb a c d e b a a b e a e a b e a b c a b d e a

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T(G,v) : the path tree of G at v vertices are paths in G starting at v edges to paths differing in one step Theorem: The matching polynomial divides the characteristic polynomial of T(G,v)

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Godsils Proof of Heilmann-Lieb T(G,v) : the path tree of G at v vertices are paths in G starting at v edges to paths differing in one step Theorem: The matching polynomial divides the characteristic polynomial of T(G,v) Is a subgraph of infinite tree, so has smaller spectral radius

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Quotients of Trees (c,d) -biregular bipartite Ramanujan as quotient of infinite (c,d) -ary tree Spectral radius For (c,d) -regular bipartite Ramanujan graphs

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Irregular Ramanujan Graphs (Greenberg-Lubotzky) Def: G is Ramanujan if its non-trivial eigenvalues have abs value less than the spectral radius of its cover Theorem: If G is bipartite and Ramanujan, then there is an infinite family of Ramanujan graphs with the same cover.

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Questions Non-bipartite Ramanujan Graphs of every degree? Efficient constructions? Explicit constructions?